The moduli spaces of hyperelliptic curves and binary forms

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Hg ∼= B2g+2. LetHg andB2g+2 denote the moduli spaces of stable hyperelliptic curves of genus g and semistable binary forms of degree 2g+2 respectively. These are certainly different compactifications of Hg = B2g+2. In fact the boundary ∆ = Hg \ Hg is a divisor whereas B2g+2 is a one point compactification of the moduli space of stable binary forms. Moreover, Hg and B2g+2 are constructed as quotients via different group actions: Hg is defined as the closure of Hg in Mg the moduli space of stable curves of genus g and this is constructed using the group PGL(6g− 5), whereas B2g+2 is constructed classically using the group SL2(C| ). It is the aim of this note to work out the relation between the spacesHg and B2g+2. The main result is the following theorem, which generalizes Theorem 5.6 of [1] where we give a proof in the special case g = 2.